The fundamental frequency of vibration of a string is given by
[tex]f= \frac{1}{2L} \sqrt{ \frac{T}{\mu} } [/tex]
where
L is the length of the string
T is the tension
[tex]\mu= \frac{m}{L} [/tex] is the linear mass density, where m is the mass of the string
Let's start by calculating the mass density of the piano wire in the problem. Its mass is
[tex]m=5.60 g=5.60 \cdot 10^{-3} kg[/tex]
and its length is L=0.600 m, so its mass density is
[tex]\mu= \frac{5.60 \cdot 10^{-3} kg}{0.600 m}=9.33 \cdot 10^{-3} kg/m [/tex]
Therefore now we have everything to calculate the fundamental frequency of the piano wire. Using T=440 N and the first equation, we find:
[tex]f= \frac{1}{2 \cdot 0.600 m} \sqrt{ \frac{440 N}{9.33 \cdot 10^{-3} kg/m} } =180.9 Hz [/tex]
